Journal of Integer Sequences, Vol. 25 (2022), Article 22.4.8

Precious Metal Sequences and Sierpiński-Type Graphs

Andreas M. Hinz
Department of Mathematics
Ludwig-Maximilians-Universität München
Theresienstraße 39
80333 Munich

Paul K. Stockmeyer
Department of Computer Science
The College of William & Mary
P. O. Box 8795
Williamsburg, VA 23187-8795


Sierpiński graphs $S_p^n$ and Sierpiński triangle graphs $\widehat{S}_p^n$ form two-parametric families of connected simple graphs which are related, for $p=3$, to the Tower of Hanoi with $n$ discs and for $n\rightarrow \infty$ to the Sierpiński triangle fractal. The vertices of minimal degree play a special role as extreme vertices in $S_p^n$ and primitive vertices in $\widehat{S}_p^n$. The key concept of this note is that of an $m$-key vertex whose distance to one of the extreme or primitive vertices, respectively, is $m$ times the distance to another one. The number of such vertices and the distances occurring lead to integer sequences with respect to parameter $n$ like, e.g., the Fibonacci sequence (golden) for $p=3$ and the Pell sequence (silver) for $p=4$. The elements of most of these sequences form self-generating sets. We discuss the cases $m=1,2,3,4$ in detail.

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(Concerned with sequences A000045 A000129 A000225 A000975 A001045 A002450 A002620 A003754 A004526 A005578 A006498 A023758 A048654 A052499 A070550 A089928 A089931 A097083 A181666 A182512 A247648 A353578 A353579 A353580 A353581 A353582.)

Received February 9 2022; revised versions received May 13 2022; June 9 2022. Published in Journal of Integer Sequences, June 10 2022.

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